高指标鞍点动力学的半隐式方法及其在构建解景观中的应用
Semi-implicit method of high-index saddle dynamics and application to construct solution landscape
AI总结：
针对高指标鞍点动力学半隐式格式数值分析的难点，研究通过技术拆分等方法证明其误差估计，分析求解器收敛率，经实验验证该格式在构建复杂系统解景观时的高效性与准确性。
摘要：
我们分析了高指标鞍点动力学的半隐式格式，该格式为寻找任意指标鞍点及构建解景观提供了一种强有力的数值方法。与鞍点动力学的显式格式相比，半隐式离散放宽了步长限制并加快了收敛速度，但相应的数值分析相较于显式格式面临新的困难。具体而言，由于半隐式处理，每个时间步的特征向量正交性质无法被充分利用，且特征向量的计算与正交化过程相互耦合，这进一步增加了数值分析的复杂度。我们通过技术拆分和多变量循环归纳过程等方法解决了这些问题，证明了半隐式格式的误差估计。我们进一步分析了用于求解半隐式系统的广义最小残差求解器的收敛速率。我们开展了大量数值实验，证实了半隐式格式在构建复杂系统解景观方面的效率与准确性。
英文摘要：
We analyze the semi-implicit scheme of high-index saddle dynamics, which provides a powerful numerical method for finding the any-index saddle points and constructing the solution landscape. Compared with the explicit schemes of saddle dynamics, the semi-implicit discretization relaxes the step size and accelerates the convergence, but the corresponding numerical analysis encounters new difficulties compared to the explicit scheme. Specifically, the orthonormal property of the eigenvectors at each time step could not be fully employed due to the semi-implicit treatment, and computations of the eigenvectors are coupled with the orthonormalization procedure, which further complicates the numerical analysis. We address these issues to prove error estimates of the semi-implicit scheme via, e.g. technical splittings and multi-variable circulating induction procedure. We further analyze the convergence rate of the generalized minimum residual solver for solving the semi-implicit system. Extensive numerical experiments are carried out to substantiate the efficiency and accuracy of the semi-implicit scheme in constructing solution landscapes of complex systems.
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